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Primes whose base-9 representation is a square in base 10.
+10
4
23, 73, 509, 643, 751, 1093, 1153, 1481, 1783, 2029, 6067, 8011, 8933, 9613, 16061, 24763, 30803, 44203, 64633, 74521, 82723, 88003, 94033, 103171, 106619, 118669, 126541, 133321, 154571, 158293, 163561, 165883, 179821, 197339, 202393, 207643, 225523, 234733, 264083, 268003, 284593, 288661, 324931
EXAMPLE
73 in base 9 is 81, and 81 = 9^2 in base 10.
MATHEMATICA
Select[Prime@ Range@ 28000, IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 9] &] (* Michael De Vlieger, Jan 24 2016 *)
PROG
(Python)
from sympy import isprime
A267821_list = [int(d, 9) for d in (str(i**2) for i in range(1, 10**6)) if max(d) < '9' and isprime(int(d, 9))] # Chai Wah Wu, Feb 22 2016
Primes whose base-5 representation is a square when read in base 10.
+10
2
89, 139, 4801, 6829, 9601, 48661, 305551, 539849, 867001, 3503449, 3527071, 6063839, 13912039, 15933439, 18100351, 18319321, 20366399, 22849121, 26239321, 46724801, 63565049, 78057911, 95802799, 100294871, 102369439, 107474911, 139769449, 141118079, 144829879, 159468079, 161483801
EXAMPLE
89 in base 5 is 324, and 324 = 18^2 in base 10.
MATHEMATICA
Select[Prime@ Range[10^6], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 5] &] (* Michael De Vlieger, Jan 24 2016 *)
PROG
(Python)
from sympy import isprime
A267819_list = [int(d, 5) for d in (str(i**2) for i in range(1, 10**6)) if max(d) < '5' and isprime(int(d, 5))] # Chai Wah Wu, Feb 22 2016
Primes whose base-6 representation is a square when read in base 10.
+10
2
17, 89, 409, 449, 577, 953, 7649, 12401, 14593, 23689, 28393, 29033, 30689, 36809, 40153, 46457, 47969, 54577, 56393, 59273, 65033, 72649, 78713, 92033, 98953, 106033, 141353, 146249, 150209, 169657, 176489, 199889, 205433, 221849, 235273, 238481, 245033, 258569, 381841, 394633, 441193, 529273
EXAMPLE
17 in base 6 is 25, and 25 = 5^2 in base 10.
MATHEMATICA
Select[Prime@ Range@ 45000, IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 6] &] (* Michael De Vlieger, Jan 24 2016 *)
PROG
(Python)
from sympy import isprime
A267820_list = [int(d, 6) for d in (str(i**2) for i in range(1, 10**6)) if max(d) < '6' and isprime(int(d, 6))] # Chai Wah Wu, Feb 22 2016
Primes whose base-4 representation is a square when read in base 10.
+10
1
673, 1165921, 6228961, 53378329, 128818873, 805379161, 926711449, 1542173569, 2459671993, 8365652929, 14232980953, 26484614209, 30755887321, 31937961241, 37968797689, 78180076129, 80974979329, 97835803129, 168394504129, 314911422913, 410879716321, 529074406681
EXAMPLE
673 written in base 4 is 22201, and 22201 in base 10 is 149^2.
MATHEMATICA
Select[Prime@ Range[10^6], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 4] &] (* Michael De Vlieger, Jan 24 2016 *)
PROG
(PARI) is(n, b=4, c=10)=issquare(subst(Pol(digits(n, b)), x, c))
lista(nn) = forprime(p=2, nn, if (is(p), print1(p, ", "))) \\ Michel Marcus, Jan 24 2016 (Python)
from sympy import isprime
A267818_list = [int(d, 4) for d in (str(i**2) for i in range(1, 10**6)) if max(d) < '4' and isprime(int(d, 4))] # Chai Wah Wu, Feb 22 2016
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